Advertisements
Advertisements
प्रश्न
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
विकल्प
4
6
8
10
Advertisements
उत्तर
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to 6.
Explanation:
Sn = `n/2 [2a + (n - 1)d]`
∴ S2n = `(2n)/2 [2a + (2n - 1)d]`
3Sn = `(3n)/2 [2a + (3n - 1)d]`
We have S2n = 3 · Sn
⇒ `(2n)/2 [2a + (2n - 1)d] = 3 * n/2 [2a + (n - 1)d]`
⇒ 2[2a + (2n – 1)d] = 3[2a + (n – 1)d]
⇒ 4a + (4n – 2)d = 6a + (3n – 3)d
⇒ 6a + (3n – 3)d – 4a – (4n – 2)d = 0
⇒ 2a + (3n – 3 – 4n + 2)d = 0
⇒ 2a + (– n – 1)d = 0
⇒ 2a – (n + 1)d = 0
⇒ 2a = (n + 1)d ....(i)
Now S3n: Sn = `(3n)/2 [2a + (3n - 1)d] : n/2 [2a + (n - 1)d]`
= `((3n)/2 [2a + (3n - 1)d])/(n/2 [2a + (n - 1)d])`
= `(3[2a + (3n - 1)d])/(2a + (n - 1)d)`
= `(3[(n + 1)d + (3n - 1)d])/((n + 1)d + (n - 1)d)`
= `(3d[n + 1 + 3n - 1])/(d(n + 1 + n - 1))`
= `(3[4n])/(2n)`
= 6
APPEARS IN
संबंधित प्रश्न
Find the sum of odd integers from 1 to 2001.
In an A.P, the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20th term is –112.
The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon.
Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)
Let < an > be a sequence. Write the first five term in the following:
a1 = a2 = 2, an = an − 1 − 1, n > 2
Which term of the sequence 24, \[23\frac{1}{4,} 22\frac{1}{2,} 21\frac{3}{4}\]....... is the first negative term?
How many terms are there in the A.P. 7, 10, 13, ... 43 ?
The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.
Find the 12th term from the following arithmetic progression:
1, 4, 7, 10, ..., 88
The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.
Find the sum of the following arithmetic progression :
1, 3, 5, 7, ... to 12 terms
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
Solve:
25 + 22 + 19 + 16 + ... + x = 115
Solve:
1 + 4 + 7 + 10 + ... + x = 590.
Find the sum of n terms of the A.P. whose kth terms is 5k + 1.
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
How many terms of the A.P. −6, \[- \frac{11}{2}\], −5, ... are needed to give the sum −25?
In an A.P. the first term is 2 and the sum of the first five terms is one fourth of the next five terms. Show that 20th term is −112.
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18th terms.
A man saves Rs 32 during the first year. Rs 36 in the second year and in this way he increases his savings by Rs 4 every year. Find in what time his saving will be Rs 200.
A manufacturer of radio sets produced 600 units in the third year and 700 units in the seventh year. Assuming that the product increases uniformly by a fixed number every year, find (i) the production in the first year (ii) the total product in 7 years and (iii) the product in the 10th year.
We know that the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.
If the sums of n terms of two arithmetic progressions are in the ratio 2n + 5 : 3n + 4, then write the ratio of their m th terms.
Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
In the arithmetic progression whose common difference is non-zero, the sum of first 3 n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2 n terms to the next 2 nterms is
If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are
The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by \[\frac{l^2 - a^2}{k - (l + a)}\] , then k =
Mark the correct alternative in the following question:
If in an A.P., the pth term is q and (p + q)th term is zero, then the qth term is
Mark the correct alternative in the following question:
Let Sn denote the sum of first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to
If second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.
The first term of an A.P. is a, the second term is b and the last term is c. Show that the sum of the A.P. is `((b + c - 2a)(c + a))/(2(b - a))`.
A man saved Rs 66000 in 20 years. In each succeeding year after the first year he saved Rs 200 more than what he saved in the previous year. How much did he save in the first year?
Let 3, 6, 9, 12 ....... upto 78 terms and 5, 9, 13, 17 ...... upto 59 be two series. Then, the sum of the terms common to both the series is equal to ______.
If a1, a2, a3, .......... are an A.P. such that a1 + a5 + a10 + a15 + a20 + a24 = 225, then a1 + a2 + a3 + ...... + a23 + a24 is equal to ______.
